where the exchange integral \(J\)  vanishes exponentially as a function of
distance between the local moments on an interatomic length scale. The
magneto- and electro-dipolar interactions have the same angle and distance
dependence, but different prefactors [1]}. The interaction energy
of two parallel magnetic moments with \(\mathbf {M}_{i}=\mu _{B}\)  at a distance
\(r\)  is
\(F_{M}=\frac{\mu _{0}\mu _{B}^{2}}{4\pi }\frac{1}{r^{3}}\sim \frac{\alpha ^{2}}{4\pi }\left( \frac{a_{B}}{r}\right) ^{3},\) 
